Fundamental Methods of Mathematical Economics (COLLEGE IE (REPRINTS))

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Boston Burr Ridge, IL Dubuque, IA Madison, WI New York San Francisco St. Louis Bangkok Bogotá Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan Montreal New Delhi Santiago Seoul Singapore Sydney Taipei Toronto which pertain, respectively, to the equilibrium of a market model and the equilibrium of the national-income model in its simplest form. Similarly, an optimization model either derives or applies one or more optimization conditions. One such condition that comes easily to mind is the condition MC = MR Inflation and Unemployment in Discrete Time 581 The Model 581 The Difference Equation in p 582 The Time Path of p 583 The Analysis of U 584 The Long-Run Phillips Relation 585 Exercise 18.3 585 Alternative Terminal Conditions 639 Fixed Terminal Point 639 Horizontal Terminal Line 639 Truncated Vertical Terminal Line 639 Truncated Horizontal Terminal Line 640 Exercise 20.2 643

Fundamental methods of mathematical economics - Archive.org

behavioral equations can be used to describe the general institutional setting of a model, including the technological (e.g., production function) and legal (e.g., tax structure) aspects. Before a behavioral equation can be written, however, it is always necessary to adopt definite assumptions regarding the behavior pattern of the variable in question. Consider the two cost functions C = 75 + 10Q (2.1) C = 110 + Q 2 The common property of all fractional numbers is that each is expressible as a ratio of two integers. Any number that can be expressed as a ratio of two integers is called a rational number. But integers themselves are also rational, because any integer n can be considered as the ratio n/1. The set of all integers and the set of all fractions together form the set of all rational numbers. An alternative defining characteristic of a rational number is that it is expressible as either a terminating decimal (e.g., 14 = 0.25) or a repeating decimal (e.g., 1 = 0.3333 . . .), where some number or series of numbers to the right of the decimal point 3 is repeated indefinitely. Once the notion of rational numbers is used, there naturally arises the concept of irrational numbers—numbers √ that cannot be expressed as ratios of a pair of integers. One example is the number 2 = 1.4142 . . . , which is a nonrepeating, nonterminating decimal. Another is the special constant π = 3.1415 . . . (representing the ratio of the circumference of any circle to its diameter), which is again a nonrepeating, nonterminating decimal, as is characteristic of all irrational numbers. Each irrational number, if placed on a ruler, would fall between two rational numbers, so that, just as the fractions fill in the gaps between the integers on a ruler, the irrational numbers fill in the gaps between rational numbers. The result of this filling-in process is a continuum of numbers, all of which are so-called real numbers. This continuum constitutes the set of all real numbers, which is often denoted by the symbol R. When the set R is displayed on a straight line (an extended ruler), we refer to the line as the real line. In Fig. 2.1 are listed (in the order discussed) all the number sets, arranged in relationship to one another. If we read from bottom to top, however, we find in effect a classificatory scheme in which the set of real numbers is broken down into its component and subcomponent number sets. This figure therefore is a summary of the structure of the real-number system. Real numbers are all we need for the first 15 chapters of this book, but they are not the only numbers used in mathematics. In fact, the reason for the term real is that there are also “imaginary” numbers, which have to do with the square roots of negative numbers. That concept will be discussed later, in Chap. 16. Natural Exponential Functions and the Problem of Growth 260 The Number e 260 An Economic Interpretation of e 262 Interest Compounding and the Function Ae rt 262 Instantaneous Rate of Growth 263 Continuous versus Discrete Growth 265 Discounting and Negative Growth 266 Exercise 10.2 267 As a third possible type of set relationship, two sets may have no elements in common at all. In that case, the two sets are said to be disjoint. For example, the set of all positive integers and the set of all negative integers are mutually exclusive; thus they are disjoint sets. A fourth type of relationship occurs when two sets have some elements in common but some elements peculiar to each. In that event, the two sets are neither equal nor disjoint; also, neither set is a subset of the other. very good math preparation book for econ major. It could be used to undergraduate text, or graduate review book, since the level is a bit lower.Nonlinear Differential Equations of the First Order and First Degree 492 Exact Differential Equations 492 Separable Variables 492 Equations Reducible to the Linear Form 493 Exercise 15.5 495 Chapter 1 The Nature of Mathematical Economics 2 1.1 Mathematical versus Nonmathematical Economics 2 1.2 Mathematical Economics versus Econometrics 4

Fundamental Mathematical Economic solutions - Studocu Chiang Fundamental Mathematical Economic solutions - Studocu

Constant-Function Rule 148 Power-Function Rule 149 Power-Function Rule Generalized 151 Exercise 7.1 152 Chapter 10 Exponential and Logarithmic Functions 255 10.1 The Nature of Exponential Functions 256 Simple Exponential Function 256 Graphical Form 256 Generalized Exponential Function 257 A Preferred Base 259 Exercise 10.1 260

The Interaction of Inflation and Unemployment 532 The Phillips Relation 532 The Expectations-Augmented Phillips Relation 533 The Feedback from Inflation to Unemployment 534 The Time Path of π 534 Exercise 16.5 537 Differential Equations with a Variable Term 538 Method of Undetermined Coefficients 538 A Modification 539 Exercise 16.6 540 Partial Differentiation 165 Partial Derivatives 165 Techniques of Partial Differentiation 166 Geometric Interpretation of Partial Derivatives 167 Gradient Vector 168 Exercise 7.4 169

Fundamental methods of mathematical economics - Open Library Fundamental methods of mathematical economics - Open Library

The Derivative and the Slope of a Curve 128 6.4 The Concept of Limit 129 Left-Side Limit and Right-Side Limit 129 Graphical Illustrations 130 Evaluation of a Limit 131 Formal View of the Limit Concept 133 Exercise 6.4 135 Chapter 15 Continuous Time: First-Order Differential Equations 475 15.1 First-Order Linear Differential Equations with Constant Coefficient and Constant Term 475 The Homogeneous Case 476 The Nonhomogeneous Case 476 Verification of the Solution 478 Exercise 15.1 479Chapter 4 Linear Models and Matrix Algebra 48 4.1 Matrices and Vectors 49 Matrices as Arrays 49 Vectors as Special Matrices 50 Exercise 4.1 51 Least-Cost Combination of Inputs 390 First-Order Condition 390 Second-Order Condition 392 The Expansion Path 392 Homothetic Functions 394 Elasticity of Substitution 396 CES Production Function 397 Cobb-Douglas Function as a Special Case of the CES Function 399 Exercise 12.7 401 AB = ⎡ ⎣(4×3) + (7×2) (4×8) + (7×6) (4×5) + (7×7) (9×3) + (1×2) (9×8) + (1×6) (9×5) + (1×7) ⎤ ⎦ = ⎡ ⎣26 74 69 29 78 52 ⎤ ⎦ The Nature of Mathematical Economics Mathematical economics is not a distinct branch of economics in the sense that public finance or international trade is. Rather, it is an approach to economic analysis, in which the economist makes use of mathematical symbols in the statement of the problem and also draws upon known mathematical theorems to aid in reasoning. As far as the specific subject matter of analysis goes, it can be micro- or macroeconomic theory, public finance, urban economics, or what not. Using the term mathematical economics in the broadest possible sense, one may very well say that every elementary textbook of economics today exemplifies mathematical economics insofar as geometrical methods are frequently utilized to derive theoretical results. More commonly, however, mathematical economics is reserved to describe cases employing mathematical techniques beyond simple geometry, such as matrix algebra, differential and integral calculus, differential equations, difference equations, etc. It is the purpose of this book to introduce the reader to the most fundamental aspects of these mathematical methods—those encountered daily in the current economic literature. Nonlinear Difference Equations— The Qualitative-Graphic Approach 562 Phase Diagram 562 Types of Time Path 564 A Market with a Price Ceiling 565 Exercise 17.6 567

Fundamental Methods of Mathematical Economics - McGraw Hill

Elegant Yet Lucid Writing Style: Chiang?s strength is the eloquence of the writing and the manner in which it is developed. While the content of the text can be difficult, it is understandable. However, the author is very good at organizing material. Each part is started with an intuitive instruction and is closed with conclusion part which states the limitation with a certain method. So although this is a math book, the words really account, which really helps you to understand. Relationships between Sets When two sets are compared with each other, several possible kinds of relationship may be observed. If two sets S1 and S2 happen to contain identical elements, S1 = {2, 7, a, f }

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Quadratic Forms—An Excursion 301 Second-Order Total Differential as a Quadratic Form 301 Positive and Negative Definiteness 302 Determinantal Test for Sign Definiteness 302 Three-Variable Quadratic Forms 305 n-Variable Quadratic Forms 307 Characteristic-Root Test for Sign Definiteness 307 Exercise 11.3 312 Domar Growth Model 471 The Framework 471 Finding the Solution 472 The Razor’s Edge 473 Exercise 14.6 474 Second-Derivative Test 233 Necessary versus Sufficient Conditions 234 Conditions for Profit Maximization 235 Coefficients of a Cubic Total-Cost Function 238 Upward-Sloping Marginal-Revenue Curve 240 Exercise 9.4 241